Internal problem ID [10060]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2. Equations of
the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 66.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime } y-a \left (\frac {n +2}{n}+b \,x^{n}\right ) y+\frac {a^{2} x \left (\frac {n +1}{n}+b \,x^{n}\right )}{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 202
dsolve(y(x)*diff(y(x),x)-a*((n+2)/n+b*x^n)*y(x)=-a^2/n*x*((n+1)/n+b*x^n),y(x), singsol=all)
\[ c_{1}-\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, \left (\int _{}^{\frac {2 \arctan \left (\frac {2 x^{n +1} a b n +x a n -n^{2} y \relax (x )+a x -n y \relax (x )}{n \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, \left (a x -n y \relax (x )\right )}\right )}{\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}}\tan \left (\frac {\textit {\_a} \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}{2}\right ) {\mathrm e}^{-\textit {\_a}}d \textit {\_a} \right ) n +\left (-2 b \,x^{n} n -n -1\right ) {\mathrm e}^{-\frac {2 \arctan \left (\frac {2 x^{n +1} a b n +x a n -n^{2} y \relax (x )+a x -n y \relax (x )}{n \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, \left (a x -n y \relax (x )\right )}\right )}{\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-a*((n+2)/n+b*x^n)*y[x]==-a^2/n*x*((n+1)/n+b*x^n),y[x],x,IncludeSingularSolutions -> True]
Not solved